In this section, we provide mathematical analysis to understand NFM. We begin with formulating NFM in the framework of vicinal risk minimization and interpreting NFM as a stochastic learning strategy in Subsection REF . Next, we study NFM via the lens of implicit regularization in Subsection REF . Our key contribution is Theorem REF , which shows that minimizing the NFM loss function is approximately equivalent to minimizing a sum of the original loss and feature-dependent regularizers, amplifying the regularizing effects of manifold mixup and noise injection according to the mixing and noise levels. In Subsection REF , we focus on demonstrating how NFM can enhance model robustness via the lens of distributionally robust optimization. The key result of Theorem REF  shows that NFM loss is approximately the upper bound on a regularized version of an adversarial loss, and thus training with NFM not only improves robustness but can also mitigate robust over-fitting, a dominant phenomenon where the robust test accuracy starts to decrease during training [1]}.
